Deriving Compound Angle Identities: Additional Trigonometric Proofs

These workings use compound angle identities to derive double angle, triple angle, and related trigonometric identities.

Compound Angles, Extras

1. Deriving sin(2θ)

sin(θ + θ) = sinθ cosθ + cosθ sinθ

= 2sinθ cosθ

= sin(2θ)

2. Deriving cos(2θ)

cos(θ + θ) = cosθ cosθ - sinθ sinθ

= cos2θ - sin2θ

= cos(2θ)

3. Deriving cos(2θ) = 2cos²θ - 1

cos(2θ) = cos2θ - sin2θ

= cos2θ - (1 - cos2θ)

= cos2θ - 1 + cos2θ

= 2cos2θ - 1

4. Deriving cos(2θ) = 1 - 2sin²θ

cos(2θ) = cos2θ - sin2θ

= 1 - sin2θ - sin2θ

= 1 - 2sin2θ

5. Deriving sin(3θ)

sin(2θ + θ) = sin(2θ)cosθ + cos(2θ)sinθ

= 2sinθ cosθ cosθ + (1 - 2sin2θ)sinθ

= 2sinθ cos2θ + sinθ - 2sin3θ

= sinθ(2cos2θ + 1) - 2sin3θ

= sinθ(2(1 - sin2θ) + 1) - 2sin3θ

= sinθ(2 - 2sin2θ + 1) - 2sin3θ

= sinθ(3 - 2sin2θ) - 2sin3θ

= 3sinθ - 2sin3θ - 2sin3θ

= 3sinθ - 4sin3θ

6. Deriving cos(3θ)

cos(2θ + θ) = cos2θ cosθ - sin2θ sinθ

= (2cos2θ - 1)cosθ - (2sinθ cosθ)sinθ

= 2cos3θ - cosθ - 2sin2θ cosθ

= 2cos3θ - cosθ - 2(1 - cos2θ)cosθ

= 2cos3θ - cosθ - 2cosθ + 2cos3θ

= 4cos3θ - 3cosθ

= cos3θ

7. Deriving cos²θ from cos(2θ)

1/2(1 + cos2θ)

= 1/2(1 + 2cos2θ - 1)

= 1/2 · 2cos2θ

= cos2θ

8. Deriving sin²θ from cos(2θ)

1/2(1 - cos2θ)

= 1/2(1 - (1 - 2sin2θ))

= 1/2(1 - 1 + 2sin2θ)

= 1/2 · 2sin2θ

= sin2θ

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